Subgroups — #41006366

Order 26973.

256 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#15688510	C₃	1	3	yes	maximal, complemented
2	C₉×C₃²⋊(C₃₇⋊C₃)	C₃	1	1	yes	maximal, characteristic
3	C₃²⋊#11758601	C₃	1	2	yes	maximal, complemented
4	#22582465	C₃	1	6	yes	maximal, complemented
5	C₃²×C₂₇×C₃₇	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃×C₃²⋊(C₃₇⋊C₃)	C₉	1	1	yes	characteristic
7	C₃×C₉×C₃₇⋊C₃	C₃²	1	3	yes	—
8	#15688510	—	3	9	no	—
9	C₃×#11758601	C₃²	1	6	yes	—
10	C₃²⋊#66524	C₉	1	2	yes	—
11	#15688502	C₉	1	6	yes	—
12	C₃²×C₉×C₃₇	C₃²	1	1	yes	characteristic
13	C₃×C₂₇×C₃₇	—	3	3	no	—
14	C₃×C₂₇×C₃₇	C₃²	1	2	yes	—
15	C₃×C₂₇×C₃₇	C₃²	1	1	yes	characteristic
16	C₃³×C₃₇	C₃×C₉	1	1	yes	characteristic
17	C₃²×C₃₇⋊C₃	C₃×C₉	1	3	yes	—
18	C₃×C₉×C₃₇	—	3	1	no	—
19	C₃×C₉×C₃₇	—	3	2	no	—
20	C₃×C₉×C₃₇	C₃×C₉	1	2	yes	—
21	C₃×C₉×C₃₇	C₃³	1	1	yes	characteristic
22	C₃²⋊(C₃₇⋊C₃)	—	3	3	no	—
23	C₃×#66524	C₃×C₉	1	6	yes	—
24	C₉×C₃₇⋊C₃	—	3	3	no	—
25	C₉×C₃₇⋊C₃	—	3	6	no	—
26	C₂₇×C₃₇	—	3	2	no	—
27	C₂₇×C₃₇	H(3,𝔽₃)	1	3	yes	complemented
28	#11758601	—	3	6	no	—
29	C₂₇⋊H(3,𝔽₃)	—	37	1	no	maximal, Sylow
30	C₃²×C₃₇	—	3	1	no	—
31	C₃²×C₃₇	—	3	2	no	—
32	C₃²×C₃₇	C₂₇⋊C₃	1	3	yes	complemented
33	C₃²×C₃₇	C₃²×C₉	1	1	yes	characteristic
34	C₃×C₃₇⋊C₃	—	3	3	no	—
35	C₃×C₃₇⋊C₃	—	3	3	no	—
36	C₃×C₃₇⋊C₃	—	3	6	no	—
37	C₉×C₃₇	—	3	2	no	—
38	C₉×C₃₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
39	C₉×C₃₇	#737	1	2	yes	—
40	#66524	—	3	6	no	—
41	C₃×C₂₇⋊C₃	—	37	3	no	—
42	C₃²×C₂₇	C₃₇⋊C₃	1	1	yes	characteristic, complemented
43	C₉×H(3,𝔽₃)	—	37	1	no	—
44	C₃²⋊C₂₇	—	37	2	no	—
45	C₃²⋊C₂₇	—	37	6	no	—
46	C₃×C₃₇	—	3	3	no	—
47	C₃×C₃₇	C₉×H(3,𝔽₃)	1	1	yes	characteristic
48	C₃×C₃₇	C₃×C₂₇⋊C₃	1	1	yes	characteristic
49	C₃×C₃₇	#6903	1	2	yes	—
50	C₃₇⋊C₃	—	9	3	no	—
51	C₃×H(3,𝔽₃)	—	37	1	no	—
52	C₃²×C₉	—	37	3	no	—
53	C₃²×C₉	C₃×C₃₇⋊C₃	1	1	yes	characteristic
54	C₂₇⋊C₃	—	111	9	no	—
55	C₃×C₂₇	—	3	3	no	—
56	C₃×C₂₇	—	37	6	no	—
57	C₃×C₂₇	C₃×C₃₇⋊C₃	1	2	yes	—
58	C₃×C₂₇	C₃×C₃₇⋊C₃	1	1	yes	characteristic
59	C₃²⋊C₉	—	37	2	no	—
60	C₃²⋊C₉	—	37	6	no	—
61	C₃₇	C₂₇⋊H(3,𝔽₃)	1	1	yes	characteristic, complemented, Sylow
62	C₃³	—	37	3	no	—
63	C₃³	C₉×C₃₇⋊C₃	1	1	yes	characteristic
64	H(3,𝔽₃)	—	111	3	no	—
65	C₃×C₉	—	3	1	no	—
66	C₃×C₉	—	3	2	no	—
67	C₃×C₉	—	37	6	no	—
68	C₃×C₉	—	111	3	no	—
69	C₃×C₉	—	111	6	no	—
70	C₃×C₉	C₉×C₃₇⋊C₃	1	2	yes	—
71	C₃×C₉	C₃²×C₃₇⋊C₃	1	1	yes	characteristic
72	C₂₇	—	3	2	no	—
73	C₂₇	—	111	6	no	—
74	C₂₇	C₃²⋊(C₃₇⋊C₃)	1	3	yes	complemented
75	C₃²	—	3	1	no	—
76	C₃²	—	3	2	no	—
77	C₃²	—	111	3	no	—
78	C₃²	—	111	3	no	—
79	C₃²	—	111	6	no	—
80	C₃²	#15688510	1	3	yes	complemented
81	C₃²	C₃×C₉×C₃₇⋊C₃	1	1	yes	characteristic
82	C₉	—	3	2	no	—
83	C₉	—	111	6	no	—
84	C₉	C₃×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
85	C₉	#15688531	1	2	yes	—
86	C₃	—	3	3	no	—
87	C₃	—	333	3	no	—
88	C₃	C₉×C₃²⋊(C₃₇⋊C₃)	1	1	yes	characteristic
89	C₃	C₃×#15688510	1	1	yes	characteristic
90	C₃	#22582612	1	2	yes	—

	subgroup	G/H
center, Frattini	C₃×C₉	C₃²×C₃₇⋊C₃
commutator, socle	C₃²×C₃₇	C₃²×C₉
Fitting	C₃²×C₂₇×C₃₇	C₃
prime radical	C₃×C₃²⋊(C₃₇⋊C₃)	C₉
cosocle	C₃×C₉×C₃₇	C₃³
hypercenter	C₃²×C₂₇	C₃₇⋊C₃
hypocenter	C₃₇	C₂₇⋊H(3,𝔽₃)
LCS	G ▷ C₃²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₃×C₉ ◁ C₃²×C₂₇

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₃×#15688510	C₃	semidirect
3. C₃²⋊#11758601	C₃	semidirect
4. #22582465	C₃	semidirect
5. C₃²×C₂₇×C₃₇	C₃	semidirect
27. C₂₇×C₃₇	H(3,𝔽₃)	semidirect
32. C₃²×C₃₇	C₂₇⋊C₃	semidirect
42. C₃²×C₂₇	C₃₇⋊C₃	semidirect
61. C₃₇	C₂₇⋊H(3,𝔽₃)	semidirect
74. C₂₇	C₃²⋊(C₃₇⋊C₃)	semidirect
80. C₃²	#15688510	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃×#15688510	3
C₉×C₃²⋊(C₃₇⋊C₃)	3
C₃²⋊#11758601	3
#22582465	3
C₃²×C₂₇×C₃₇	3
C₂₇⋊H(3,𝔽₃)	37